She says that every human being needs to have multiple roots. As a tree is rooted to the earth, giving it stability and strength, so people need to be rooted in several things--in a community, a tradition, a people, a sense of self. We are automatically rooted in many things just by birth--for example, in our family or our country. But when the State (the government or ruling powers, as opposed to our land and people) dictates the values of humans, and people are attached to identities bought by money, this contributes to the uprootedness and alienation of the soul in the modern age. A society corrupted by wealth and power cannot provide the roots that people need. This is what Weil is talking about. You need to look for the whole passage to understand her meaning.
My thoughts: When people get uprooted from the natural things that give them identity and stability--like war refugees--it is traumatic. A deeper rooting may come through spiritual and moral values that are developed over time. When you are rooted in a strong inner connection to purpose, a sense of meaning, and faith, then being uprooted from a geographical place or even particular people may not be as traumatic.